The global conjugacy property conjecture for even Markov groups

Let $W_n$ be the iterated wreath-product group acting on the rooted binary tree of height $n$, let $K_n=\operatorname{Ker}(\pi_n:W_n\twoheadrightarrow W_{n-1})$, and let $H,G\leq W_n$. Say that $H$ is elementwise $K_n$-conjugate into $G$ if for every $h\in H$ there exists $k_h\in K_n$ such that $h^{k_h}\in G$, and that $H$ is globally $K_n$-conjugate into $G$ if there exists $k\in K_n$ such that $H^k\leq G$. The pair $(H,G)$ satisfies property $\mathcal{P}$ when these two conditions are equivalent. Let $f(x)\in\mathbb{Z}[x]$ be as in Theorem 1.1 or Theorem 1.2, and let $M_n(f)$ be the level $n$ even Markov group of $f$. Global conjugacy property conjecture. For every $n\geq 1$ and every subgroup $H\leq W_n$, the pair $(H,M_n(f))$ satisfies $\mathcal{P}$. This conjecture would connect the group-theoretic question about elementwise versus global conjugacy to the proposed containment of Galois groups in Markov groups.